Grade 10 Quadratics

By: Ravi Seeder

Introducing Quadratics

Quadratics can be used to graph the path or flight of a certain object which are represented by parabola's on a graph and the name comes from 'quad' which means square. So if an equation has 'x2' you know if it is an quadratic equation.

Table Of Contents

1. Quadratics Relations In Vertex Form: y=a(x-h)2+k

-Axis Of Symmetry (x)

-Optimal Value (y)

-Transformation (Vertical Stretch/Compression)


-Step Pattern

-Word Problems

2. Quadratic Relations In Factored Form: y=a(x-r)(x-s)

-X-intercepts and sub/isolating

-Axis of Symmetry

-Optimal Value

-What is Factoring

-Common Factoring, Factoring Perfect Squares, Factoring Different Squares, Factoring Simple and Complex Trinomials

-Word Problems

3.Quadratic Relations in Standard form: y= ax2 + bx + c

-Quadratic Formula

-Axis Of Symmetry

-Optimal Value

-Completing The Square

-Factoring back Into Factored Form

-Word Problems

4. Reflection

- A memory of mine in quadratics.

5. Connection

-How these different methods/formulas become one as quadratics

-How all relates to each other.

1. Identifying Quadratic Relations in Vertex Form: y=a(x-h)2 +k

Example: y=2(x-2)2 +1

h= Your x-intercept. (In brackets is going to be opposite form so -2 is actually +2)

k= Your y-intercept.

Your 'h' can also represent your axis of symmetry.

Your 'y' is also represented to be your optimal value.

a= Your vertical stretch/compression. NOTE: Is your 'a' is less than 1 its is going to be compression and if it is more than one its going to be a stretch. Also Note that if your 'a' is a negative number, your parabola is going to be reflecting..


Vertex: (2,1)

Max/Min: Min because your parabola is not reflecting due to 'a'

Axis of symmetry: x=2

Quadratic equation in vertex form:

Also Note: When you are isolating for 'x' y=0 and when your

5.1 Graphing Quadratic Equations in Vertex Form

-The Step Pattern.

The step pattern is a pattern that identifies your next points in each parabola.

The original step patter is 'Over 1 up 1 and Over 2 up 4'

But in quadratic relations if there is an 'a' value the step pattern changes.

The 'a' value gets multiplied with each 'up' value.

Example: y=2(x-4)2 +1

Due to the 'a' value in this relation being 2 you must multiply each 'up' with 2.

So the new step pattern becomes: Over 1 up 2 and over 2 up 8

Step Pattern

First and Second Differences

In quadratics there is also a note to make of first and second differences. First and second differences will allow a student to identify if a problem is quadratic or a problem is linear through a chart. There is going to be a difference in your X and Y but that difference between your numbers will end up being the same difference. If it is the same the first time than you have a linear relation but if its different the first time and you have to check the differences again and they are the same the second time, you have a quadratic relation
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Word Problem In Vertex Form:

Most word problems that are given in vertex form usually being with a vertex form equation given and a base story such as a ball kicked into the air follows a parabolic path described by a question h=-0.2(d-5)2 + 10. (example) And usually these word problems ask questions like whats the maximum height the object reached, or how long was the object in the air and what was the distance at the highest point for the ball. Answering these questions usually have a lot to do with the vertex and usually have a lot of sub in 0 and isolate for a certain variable. Word problems which are answered like this:

Example: h=-0.2(d-10)2 + 20

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2. Identifying Quadratic Relations in factored form: y=a(x-r)(x-s) (GRAPHING)

Example: y=-2(x-1)(x-5):

r= your first x-intercept (Remember opposite form so -2 is actually +2)

s= your second x-intercept

a= still your vertical stretch/compression. Still reflecting or directing up depending on sign.

Axis of symmetry: Number between the two x-intercepts in this case +3. (+2 & +4)

y= Plug your axis of symmetry into the regular relation as 'x' and expand to find 'y' which will than be your vertex with axis of symmetry. (x,y)


x-intercepts are: +1 and +5

vertical stretch is: -2

Axis of symmetry: +3

Finding y:?



y= 8

So therefore the vertex is (3,8)

NOTE: You always sub and isolate for variables x and y. Isolating for x, y=0 and Isolating for y, x=0

5.1 Graphing Quadratic Equations in Intercept Form

-What Is Factoring?

Factoring, which is also called Factorizing is ways and methods of finding the factors of certain numbers. These methods in this case will help us find the factors of binomials in brackets.

For an example: We use a method known as spreading the rainbow.

For y=(x+4)(x+4)

We would spread the rainbow we would multiply the first two numbers/variables in each bracket. ( so 'x' multiply by 'x' which gives us x2. Than Add the two outside numbers/variables in each bracket.( 4x plus 4x which is 8x) and than multiply the last numbers in each bracket( so 4x4 which is 16)

So what were left with is our factored equation of y=x2 + 8x + 16.

-Common Factoring

Common Factoring is when you have an expression and you have to find a Greatest Common Factor and that is placed on the outside of your new and factored expression. Also the you divide each number in the original equation by your GCF and put it in brackets. Your GCF always stays on the outside

Example: 3x+9

when you divide the numbers trying to find the GCF it ends to be 3. So GCF=3

So your new factored equation: 3(x+3)

Factoring by Grouping - Algebra Help

-Factoring Perfect Squares

Factoring Perfect Squares is basically factoring two sets of brackets which are the same. Factoring two of the same brackets into a Standard form Equation.

Example: y=(x+4)(x+4) or y=(x+4)2

So after Factoring and spreading the rainbow, the equation will be one simple equation in standard form which will be y= x2 + 8x + 16.

Algebra - Perfect Square Factoring and Square Root Property

-Factoring Difference Of Squares

Factoring Difference Of Sqaures is very similar to factoring perfect squares but this time the signs in the brackets are not the same. So due to the sign being different in the two brackets, after spreading the rainbow and expanding your left with cancelling out the middle term.

Example: y=(s-4)(s+4)

So final answer would be y= s2 - 16

Algebra - Factoring Differences of Squares

-Factoring Simple Trinomial

Factoring simple Trinomials is basically taking a standard form equation or taking 3 terms and factoring them into two sets of brackets. Whats different about this is that when you factor into brackets the two numbers without variables can be different, they will be different but when you add or subtract equal to the last term and when you multiply they equal to the third term from the original 3 term equation that was given.

For an example: x2 + 4x + 6

which will factor into (x )(x )

factoring trinomials

-Factoring Complex Trinomial

Factoring Complex Trinomials factoring a standard form equation and making sure when you spread the rainbow of your factored equation it equals up to your standard form equation to begin with. Factoring Complex Trinomials is a little different from Simple Trionmials. This has to deal with a lot of trial and error to get to your final factored form equation that equals to your standard form equation.
3.9 Complex Trinomial Factoring

-Factoring By Grouping

Factoring By Grouping is when you have an equation with four terms. You take that equation and divide it into two sets of brackets and expand within those 2 brackets to get your final standard form equation
Introductory Algebra - Factoring - 4 Terms by Grouping

Word Problem In Factored Form:

When word problems in factored form are given, the solutions usually have something to do with 2 x-intercepts which are given in the original factored form equation. The two x-intercepts will determine where the parabola will pass the points on the x-axis and the vertical stretch or compression will determine if the parabolas are facing upwards or downwards, either positive or negative 'a' value.

An example a word problem in this form and its solution would be:

Quadratic Equations - Solving Word problems by Factoring 1c

3. Quadratic Relations in Standard form: y= ax2+ bx + c

-Quadratic Formula

The quadratic formula is a method of finding the x-coordinate of the vertex when there is a standard form equation given. After you have gotten your x-coordinate you may place that x-coordinate back into your original equation and get y, than you have your vertex.

NOTE: THE 3 TERMS IN YOUR STANDARD EQUATION ARE REPRESENTED AS ABC IN ORDER. So ax2 is represented as 'a' bx is represented as 'b' and c is represented as 'c'


The quadratic formula is:

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-Example of Quadratic Formual

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-Completing The Square

Completing the square occurs when you are give a normal standard form equation. You want to identify the vertex so you have to turn that standard form equation.

Method of doing Completing the square.

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Word Problem In Standard Form:

There are a lot of different types of word problems that can be given and solved in standard form. For an example a problem where a standard form equation is given but one of the values for a variable is not given. That you must use your communication skills, do trial and error, and see what can word with your given equation as that value.

For an example: 'If you could factor the expression x2 + kx + 6, what are the possible values that k could have, explain?'

These types of standard form word problems would be answered like this:

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4. Me & Quadratics (Reflection)

When Quadratics was first introduced to us, I was not interested like I was interested when it came to Trig or our other first few units. At first I found Quadratics challenging and annoying just because the methods and graphing in such forms was very confusing. Through out the unit I began to come in for extra help and for guidance how to do these certain methods correctly. After some practice from homework and quizzes these methods and formulas become apart of my daily routine. In some aspects I still find somethings confusing but reviewing and taking the time to understand and practice results to good marks on my final tests.

- On my way to improving in Quadratics.

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5. Connections

What interests me a lot about quadratics is that somehow all these forms and relations seem to relate to each other and in some ways can be made into one form to another. When it comes to word problems or even simplifying and expanding equations, these forms seem to relate.

For an example: You have a standard form equation, which than you can break down and simplify it into turn into a factored form equation. ALSO it works the other way around as well. You can have a factored form equation, spread the rainbow and turn that into a standard form equation.

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Another cool way these forms relate is how you can turn a standard form equation to a vertex form equation by just completing the square.
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Quadratics In Real Life

Now that you have learned a great introduction to quadratics, lets think about where you can find quadratics in real life.

For an example, When you are playing basketball and shooting a hoop, the path of your shot can be represented by a quadratics path.

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If you actually think about parabola's are every where around us, in our activities, and even when we press the button to get a drink from the water fountain. Quadratics can be found in all sorts of situations and its great that some of us can notice it!
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Grade 10 Quadratics

By: Ravi Seeder